Cloth simulation for game engine doesn't get animated despite Shape Keys have been created
Maybe Unity is importing the .fbx from the .blend in wrong ways (Unity imports .blend files as .fbx under the hood).
I downloaded your .blend file, exported manually to .fbx and I can see the cloth waving in Unity3D.

I'm using Blender 2.79 and Unity3D 2017.3.0f2.

Try as follows:


In Object mode select both mesh and armature





Select File -> Export -> FBX
Disable Add Leaf Bones





Import the .fbx file into Unity


Animations in action